问题1：以下哪项是反比例的?
(i)从事工作的工人人数和完成工作的时间。
(ii)旅途所花费的时间和以相同速度行驶的距离。
(iii)收获作物的耕地面积。
(iv)固定行程所花费的时间和车辆的速度。
(v)一个国家的人口和人均土地面积。
解决方案：
(i) As the more number of workers can complete the job in less time, so, the number of workers is inversely proportional to the time to complete the job.
(ii) More the time we drive the car, more the distance it will travel, so they are directly proportional.
(iii) If we cultivate more land, we have more crop to harvest, So, they are directly proportional.
(iv) If the speed is increased, we can reach our destination in less time, So, increasing the speed will definitely decrease the time taken, So they are inversely proportional.
(v) If the population of country increases, area of land per person decreases,
(Example: if there are 2 cookies and 2 friends, they both will get 1-1, but, there are 3 friends, then with equal division all of them will get less than 1 cookie), So, they are inversely proportional.
问题2：在电视游戏节目中，1,000卢比的奖金应平均分配给获奖者。填写下表，找出给予单个获奖者的奖金是与获奖者数量成正比还是成反比?
| Number of winners (Xi) | 1 | 2 | 4 | 5 | 8 | 10 | 20 |
| Prize for each winner in (₹) (Pi) | 1,00,000 | 50,000 | … | … | … | … | … |
解决方案：
Let’s donate the number of winner as Xi and prize Pi, where i in our case can be one of
these values – (1, 2, 4, 5, 8, 10, 20).
Observing the following pattern -> X1 * P1 = X2 * P2 i.e. 1 * 1,00,000 = 2 * 50,000
which means 1,00,000 = 1,00,000 (which is true), So, we can conclude that Xi and Pi are inversely proportional.
i.e. if number of winners increases the prize money decrease and vice versa.
Now, let’s find the prize money for the dotted(…) values, in the table.
According to inverse Proportionality property (confirmed above) :-
We have :- Xi * Pi = Xj * Pj, Where i and j have values – (1,2,4,5,8,10,20)
Therefore, X4 * P4 = X2 * P2 , on right side we can write X1 * Y1 also, that will also be true in our case.
Now, We wanted to find P4, so we take our P4 on one side and other terms on other side
Therefore,
P4 = (X2 * P2) / (X4) i.e. P4 = (2 * 50,000) / (4).
Therefore, we have
P4 = (1,00,000) / (4) i.e. P4 = 25,000
Similarly:
P5 = (1,00,000) / 5 = 20,000.
P8 = (1,00,000) / 8 = 12,500.
P10 = (1,00,000) / 10 = 10,000.
P20 = (1,00,000) / 20 = 5,000.
Below is the final table we get after filling the values.
| Number of winners | 1 | 2 | 4 | 5 | 8 | 10 | 20 |
| Prize for each winner in (₹) | 1,00,000 | 50,000 | … | … | … | … | … |
问题3：雷曼(Rehman)使用轮辐制造车轮。他希望以使任何一对连续的辐条之间的角度相等的方式固定辐条。通过完成下表来帮助他。
| Number of spokes | 4 | 6 | 8 | 10 | 12 |
|
Angle between a pair of consecutive spokes |
90° | 60° | … | … | … |
(i)轮辐的数量和成对的连续轮辐之间形成的角度成反比吗?
(ii)计算带有15个辐条的车轮上一对连续辐条之间的辐条。
(iii)如果一对连续的辐条之间的角度为40°，则需要多少个辐条。
解决方案 ：
For inverse proportionality property, the condition
4 * 90° = 6 * 60° must hold.
Solving:
4 * 90° = 6 * 60° ⇒ 360° = 360°, which is true.
So, the Number of spokes and angle between a pair of consecutive spokes are inversely proportional.
Let’s denote angle between a pair of consecutive spokes for a wheel consising of 8 spokes is “a”, 10 spokes is “b” and 12 spokes is “c”.
Therefore,
8 * a = 6 * 60° (360°)
Dividing both sides by 8
⇒ a = (6 * 60) / 8
⇒ a = (360)/(8)
⇒ a = 45°.
Similarly:
b = (360)/10
⇒ b = 36°
c = (360)/12
⇒ c = 30°.
Below is the final table we get after filling the values.
| Number of spokes | 4 | 6 | 8 | 10 | 12 |
|
Angle between a pair of consecutive spokes |
90° | 60° | 45° | 36° | 30° |
As, we have created our table, let’s answer the questions one by one.
(i) We have proved this above.
(ii) Number of spokes = 15
let, angle between a pair of consecutive spokes for a wheel of consists of 15 spokes is “d“.
Therefore, by inverse proportionality property
d * 15 = 4 * 90
Dividing both sides by 15,
d = (4 * 90)/15
Solving the right-hand side,
d = 24°
(iii) angle between two consective spokes is 40°,
Let’s denote number of spokes by “n”
Therefore, by inverse proportionality property
n * 40 = 4 * 90
Dividing both sides by 40,
n = (4 * 90) / 40
Solving RHS(right hand side),
n = 9
Therefore, number of spokes is 9.
问题4：如果将一盒糖果分配给24个孩子，每个孩子将得到5个糖果。如果孩子的数量减少4个，每个将得到多少?
解决方案：
Observing the fact that, if we increase number of children, the number of Chocolates per children will decrease.
Which proves that number of children and number of chocolates per children is inversely proportional.
We are given that if there are 24 children, each will get 5 chocolate, and we have to find the number of chocolates (say n), if there are (24 – 4) i.e.
20 children.
Therefore, by inverse proportionality property :-
n * 20 = 24 * 5
Dividing both sides by 20,
n = (24 * 5) / 20
Solving RHS,
n = 6
Therefore, each child will get 6 chocolates.
问题5：一个农民有足够的食物在他的牛中饲养20只动物，持续6天。如果再增加0只动物，那么食物将持续多久?
解决方案：
Observing the fact that, if we increase the number of animals, the food will finish in less days, which proves that number of animals and days are inversely proportional.
Let’s denote the number of days to finish food if there are 10 more animals
i.e. (20 + 10) animals i.e. 30 animals as “n”.
Therefore, from inverse proportionality property:-
n * 30 = 20 * 6
divinding both sides by 30,
n = (20 * 6) / 30
Solving RHS
n = 4
Therefore, if there were 30 animals, the food will last for 4 days.
问题6：承包商估计3个人可以在4天内为Jaswinder的房屋重新布线。如果他用的是4个人而不是3个人，那么他们应该花多长时间完成工作?
解决方案：
让我们根据给定的信息创建一个表
| Number of person | 3 | 4 |
| Number of days | 4 | … |
Observing the fact that, if we increase the number of people on a job, the job will take less time to finish (i.e. less days), which proves that the number of people is inversely proportional to the days to finish the wiring.
Let’s denote the number of days to finish the wiring by 4 persons as “n”.
Therefore, from inverse proportionality property:-
n * 4 = 3 * 4
Dividing both sides by 4
n = 3
Therefore, 4 persons take 3 days to finish the wiring at Jaswinder’s house.
问题7：一批瓶子装在25个盒子里，每个盒子里有12个瓶子。如果每箱使用20瓶包装同一批次，那么将装满多少箱?
解决方案：
让我们根据给定的信息创建一个表
| Number of boxes | 25 | … |
| Number of bottles per box | 12 | 20 |
Observing the fact that, if we increase the bottles in one box, the number of boxes required will decrease, which proves that the number of
bottles is inversely proportional to the number of boxes.
Let’s denote the number of boxes for the batch where the number of bottles in each box is 20 as “n”.
Therefore, from inverse proportionality property:-
n * 20 = 25 * 12
Dividing both sides by 20,
n = (25 * 12) / 20
solving RHS,
n = 15
Therefore, Number of boxes required is 15.
问题8：一家工厂需要42台机器才能在63天内生产给定数量的物品。在54天内需要多少台机器才能生产相同数量的物品?
解决方案：
让我们根据给定的信息创建一个表
| Number of machines | 42 | … |
| Number of days | 63 | 54 |
Observing, the fact that if the number of machines is increased, the number of days to create articles will decrease, which proves that number of machines is inversely proportional to the number of days required to produce articles.
Let’s denote the number of machines required to create articles in 54 days is “n”.
Therefore, from inverse proportionality property:-
n * 54 = 42 * 63
Dividing both sides by 54,
n = (42 * 63) / 54
Solving RHS
n = 49
Therefore, the number of machines required to create articles in 54 days, is 49.
问题9：汽车以60公里/小时的速度行驶需要2个小时才能到达目的地。汽车以80 Km / h的速度行驶需要多长时间?
解决方案：
让我们根据给定的信息创建一个表
| Speed of car (Km/h) | 60 | 80 |
| Time (hour(s)) | 2 | … |
Observing the fact that if we increase the speed of car, it will take less time reach the destination, which proves that speed of car is inversely proportional to the time taken to reach the destination.
Let’s denote the time taken to reach destination when the car is traveling at speed of 80 Km/h as “n”
Therefore, by inversely proportionally property:-
n * 80 = 2 * 60
Dividing both sides by 80,
n = (2 * 60) / 80
Solving RHS,
n = 120 / 80
n = (3/2) hours.
Therefore, time taken to reach destination, when car is traveling at
Speed of 80 Km/h is (3/2) hours.
问题10：两个人可以在3天内在房子里装上新窗户。
(i)其中一个人在工作开始之前病倒了。现在工作需要多长时间?
(ii)一天将需要多少人来安装窗户?
解决方案 ：
让我们根据给定的信息创建一个表
| Number of persons | 2 | one fell ill so (2-1)= 1 person | … |
| Number of days | 3 | … | 1 |
Observing the fact, that number of persons on a job will decrease the time to finish the job. which proves Number of persons is inversely proportional to the Number of days required to fit the window in the house.
Let’s denote the number of days required to fit window when the number of persons
Working is 1 as “x”, and denote the number of person required to fit the window in 1 day as “y”.
Now, let’s solve the (i)
(i) from inverse proportionality property:-
x * 1 = 2 * 3
Solving RHS,
x = 6
Therefore, number of days required to fit window by 1 person is 6.
(ii) from inverse proportionality property:-
y * 1 = 2 * 3
Solving RHS,
y = 6
Therefore, number of person required to fit window in 1 day is 6.
问题11：一所学校每天有8节课，每节课持续45分钟。假设学校每天上课时间相同，如果学校每天有9个学期，那么每个学期有多长时间?
解决方案：
让我们根据给定的信息创建一个表
| Number of period | 8 | 9 |
| Duration of periods (minutes) | 45 minutes | … |
Observing the fact that the duration of periods will decrease if we increase the number of periods (because the total school hours are same), which proves that number of periods is inversely proportional to the duration of each period.
Let’s denote the duration of periods (in minutes), if there were 9 periods instead of 8 as “n”.
Therefore, from inverse proportionality property:-
n * 9 = 8 * 45
Dividing both sides by 9
n = (8 * 45) / 9
Solving RHS,
n = 40 minutes
Therefore, the duration of periods will be 40 minutes.